Vanishing of Tor Over Complete Intersections

TL;DR

This paper investigates conditions under which the Tor functor vanishes over complete intersection rings, extending previous results to establish new depth-related criteria for Tor vanishing.

Contribution

It provides new results linking depth properties of modules and their tensor products to the vanishing of Tor over complete intersection rings, building on prior foundational work.

Findings

Depth conditions on modules imply Tor vanishing.

New criteria for Tor vanishing over complete intersections.

Extension of previous results by Huneke, Jorgensen, Wiegand, and Dao.

Abstract

In this paper we are concerned with the vanishing of $\textnormal{Tor}$ over complete intersection rings. Building on results of C. Huneke, D. Jorgensen and R. Wiegand, and, more recently, H. Dao, we obtain new results showing that good depth properties on the $R$-modules $M$, $N$ and $M\otimes_RN$ force the vanishing of $\textnormal{Tor}^{R}_{i}(M,N)$ for all $i\geq 1$.